Research proof index

Every zero $s$ of the Riemann zeta function in the upper half-plane lies in the critical strip: if $\zeta(s)=0$ and $\operatorname{Im}s>0$, then $0\le\operatorname{Re}s\le1.$

The Riemann zeta zero-counting function has a crude polynomial bound: there is a constant $A>0$ such that, for every $T\ge2$, $|N_\zeta(T)|\le A T^{3/2}.$

Let $Z(s)$ be the project's entire zeta surrogate, obtained by removing the pole of $\zeta(s)$ at $s=1$. There is a constant $C>0$ such that, for every $R\ge1$, the total multiplicity of the zeros of $Z$ in $|s|\le R$ is bounded by $C(1+R)^{3/2}.$